{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T18:26:00Z","timestamp":1773253560839,"version":"3.50.1"},"reference-count":51,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:00:00Z","timestamp":1604102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.<\/jats:p>","DOI":"10.3390\/e22111241","type":"journal-article","created":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T21:39:56Z","timestamp":1604180396000},"page":"1241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants"],"prefix":"10.3390","volume":"22","author":[{"given":"Alexander A.","family":"Balinsky","sequence":"first","affiliation":[{"name":"School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK"}]},{"given":"Denis","family":"Blackmore","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6390-4627","authenticated-orcid":false,"given":"Rados\u0142aw","family":"Kycia","sequence":"additional","affiliation":[{"name":"Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, 31-155 Krak\u00f3w, Poland"}]},{"given":"Anatolij K.","family":"Prykarpatski","sequence":"additional","affiliation":[{"name":"Department of Physics, Mathematics and Computer Science, Cracov University of Technology, 31-155 Krak\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"530","DOI":"10.2307\/2319761","article-title":"The equation \u2202f\/\u2202dx = \u2202f\/\u2202y","volume":"82","author":"Chernoff","year":"1975","journal-title":"Am. 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